Moduli Spaces, Quivers, and Duality

University Of Chicago, Chicago IL

Investigators

Abstract

Dualities play a fundamental role in both mathematics and physics. A duality provides a symmetric correspondence between various objects of a pair of typically quite different theories. This is important because a difficult problem in one theory often corresponds, under the duality, to a simpler problem in the dual theory. Two important and challenging dualities are Langlands duality, which has origins in number theory, and mirror symmetry, which was first discovered by theoretical physicists and is playing an increasingly important role in mathematics as well. Neither of these two dualities is fully understood. Both of these dualities will be studied in this project. It is anticipated that the work will establish new and unexpected links between seemingly unrelated topics. This research project will explore a new approach to the study of exponential sums and integrals over moduli spaces associated to a wide class of categories. In the case of the moduli space of quiver representations, the project aims to develop a new proof of the Kac conjecture as well as a new formula, which involves a potential and which is closely related to the Donaldson-Thomas invariants introduced by Kontsevich and Soibelman. The project will also investigate a similar counting problem for the moduli space of parabolic (vector) bundles on a projective algebraic curve with punctures.